**About the course**

The word ‘Wavelet’ refers to a little wave. Wavelets are functions
designed to be considerably localized in both time and frequency
domains. There are many practical situations in which one needs to
analyze the signal simultaneously in both the time and frequency
domains, for example, in audio processing, image enhancement, analysis
and processing, geophysics and in biomedical engineering. Such analysis
requires the engineer and researcher to deal with such functions that
have an inherent ability to localize as much as possible in the two
domains simultaneously.This poses a fundamental challenge because such a
simultaneous localization is ultimately restricted by the uncertainty
principle for signal processing. Wavelet transforms have recently gained
popularity in those fields where Fourier analysis has been
traditionally used because of the property, which enables them to
capture local signal behavior. The whole idea of wavelets manifests
itself differently in many different disciplines, although the basic
principles remain the same.
Aim of the course is to introduce the idea of wavelets, filter banks and
time-frequency analysis. Haar wavelets have been introduced as an
important tool in the analysis of signal at various level of resolution.
Keeping this goal in mind, idea of representing a general finite energy
signal by a piecewise constant representation is developed. Concept of
ladder of subspaces, in particular the notion of ‘approximation’ and
‘Incremental’ subspaces is introduced. Connection between wavelet
analysis and Multirate digital systems have been emphasized, which
brings us to the need of establishing equivalence of sequences and
finite energy signals and this goal is achieved by the application of
basic ideas from linear algebra. Then the relation between wavelets and
Multirate filter banks, from the point of view of implementation is
explained.

Core/Elective,for UG/PG

Core,Both

Exposure to Signals and Systems and some basic Engineering Mathematics

Prof. Vikram M. Gadre is currently a Professor at Department of Electrical Engineering, IIT Bombay. He received his Undergraduate degree, along with President’s Gold Medal for cumulative performance during his B.Tech, from IIT Delhi in 1989. He received his PhD degree in Electrical Engineering from Indian Institute of Technology, Delhi in 1994.

His research interests are Communication and signal processing, with emphasis on multiresolution and multi-rate signal processing, especially wavelets and filter banks: theory and applications. He is known for his unique way of teaching for which he received Award for Excellence in Teaching four times from IIT Bombay.

His other recognitions and awards include: S.S.I. Varshney Award from the Systems Society of India (S.S.I) (2011), IIT Bombay Research Paper Award (2008), Felicitation from Society for Cancer Research and Communication (SCRAC), India (2006), Sixth SVC Aiya Memorial Award for Telecom Education from IETE Pune Centre (2005), 11th IETE Prof K Sreenivasan Memorial Award(2004), INAE Young Engineer Award from the Indian National Academy of Engineers (2001), Student Journal Award of the IETE(1994), Adarsh Ratna Bhagat Award from National Service Scheme,IIT Delhi(1992)

**Course layout**

**Week 1**

Module 1

Lecture 1. Introduction

Lecture 2. Origin of Wavelets

Lecture 3. Haar Wavelet

Module 2

Lecture 1. Dyadic Wavelet

Lecture 2. Dilates and Translates of Haar Wavelets

Lecture 3.L^{2} Norm of a Function

Module 3

Lecture 1.Piecewise Constant Representation of a Function

Lecture 2.Ladder of Subspaces

Lecture 3.Scaling Function for Haar Wavelet Demo:

Demonstration: Piecewise constant approximation of functions

**Week 2**

Module 4

Lecture 1. Vector Representation of Sequences

Lecture 2. Properties of Norm

Lecture 3. Parseval's Theorem

Module 5

Lecture 1. Equivalence of sequences and functions

Lecture 2. Angle between Functions & their Decomposition

Demo: Additional Information on Direct-Sum

Module 6

Lecture 1.Introduction to filter banks

Lecture 2.Haar Analysis Filter Bank in Z-domain

Lecture 3.Haar Synthesis Filter Bank in Z-domain.

Module 7

Lecture 1.Moving from Z-domain to frequency domain

Lecture 2.Frequency Response of Haar Analysis Low pass Filter bank

Lecture 3.Frequency Response of Haar Analysis High pass Filter bank

**Week 3**

Module 8

Lecture 1.Ideal two-band filter bank

Lecture 2.Disqualification of Ideal filter bank

Lecture 3.Realizable two-band filter bank

Module 9

Lecture 1.Relating Fourier transform of scaling function to filter bank

Lecture 2.Fourier transform of scaling function

Lecture 3.Construction of scaling and wavelet functions from filter bank

Demo: Demonstration: Constructing scaling and wavelet functions.

Module 10

Lecture 1.Introduction to upsampling and down sampling as Multirate operations

Lecture 2.Up sampling by a general factor M- a Z-domain analysis.

Lecture 3.Down sampling by a general factor M- a Z-domain analysis.

**Week 4**

Module 11

Lecture 1.Z
domain analysis of 2 channel filter bank.

Lecture 2.Effect of X (-Z) in time domain and aliasing.

Lecture 3.Consequences
of aliasing and simple approach to avoid it

Module 12

Lecture 1.Revisiting aliasing and the Idea of perfect reconstruction

Lecture 2.Applying perfect reconstruction and alias cancellation on Haar MRA

Lecture 3.Introduction to Daubechies family of MRA.

**Week 5**

Module 13

Lecture 1.Power
Complementarity of low pass filter

Lecture 2.Applying perfect reconstruction condition to obtain filter coefficient

Module 14

Lecture 1.Effect of minimum phase requirement on filter coefficients

Lecture 2.Building compactly supported scaling functions

Lecture 3.Second member of Daubechies family.

**Week 6**

Module 15

Lecture 1.Fourier
transform analysis of Haar scaling and Wavelet functions

Lecture 2.Revisiting Fourier Transform and Parseval's theorem

Lecture 3.Transform Analysis of Haar Wavelet function

Module 16

Lecture 1.Nature of Haar scaling and Wavelet functions in frequency domain

Lecture 2.The Idea of Time-Frequency Resolution.

Lecture 3.Some thoughts on Ideal time- frequency domain behavior

Week 7

Module 17

Lecture 1.Defining
Probability Density function

Lecture 2.Defining Mean, Variance and “containment in a given domain”

Lecture 3.Example: Haar Scaling function

Lecture 4.Variance from a slightly different perspective

Module 18

Lecture 1.Signal transformations: effect on mean and variance

Lecture 2.Time-Bandwidth product and its properties.

Lecture 3.Simplification of Time-Bandwidth formulae

Module 19

Lecture 1. Introduction

Lecture 2.Evaluation of Time-Bandwidth product

Lecture 3.Optimal function in the sense of Time-Bandwidth product

**Week 8**

Module 20

Lecture 1.Discontent
with the “Optimal function”.

Lecture 2.Journey from infinite to finite Time-Bandwidth product of Haar scaling function

Lecture 3.More insights about Time-Bandwidth product

Lecture 4.Time-frequency plane

Lecture 5.Tiling the Time-frequency plane

Module 21

Lecture 1.STFT: Conditions for valid windows

Lecture 2.STFT: Time domain and frequency domain formulations.

Lecture 3.STFT: Duality in the interpretations

Lecture 4.Continuous Wavelet Transform (CWT)

Conclusive Remarks and Future Prospects

**Suggested Reading**

1. Michael W. Frazier, "An Introduction to Wavelets through Linear Algebra”, Springer, 1999.

2. Stephane Mallat, "A Wavelet Tour of Signal Processing", Academic Press, Elsevier, 1998, 1999, Second Edition.

3. http://nptel.ac.in/courses/117101001/: The lecture series on Wavelets and Multirate Digital Signal Processing created by Prof. Vikram M. Gadre in NPTEL.

4. Barbara Burke Hubbard, "The World according to Wavelets - A Story of a Mathematical Technique in the making", Second edition, Universities Press (Private) India Limited 2003.

5. P.P. Vaidyanathan, "Multirate Systems and Filter Banks", Pearson Education, Low Price Edition.

Certification exam:

Module 1

Lecture 1. Introduction

Lecture 2. Origin of Wavelets

Lecture 3. Haar Wavelet

Module 2

Lecture 1. Dyadic Wavelet

Lecture 2. Dilates and Translates of Haar Wavelets

Lecture 3.L

Module 3

Lecture 1.Piecewise Constant Representation of a Function

Lecture 2.Ladder of Subspaces

Lecture 3.Scaling Function for Haar Wavelet Demo:

Demonstration: Piecewise constant approximation of functions

Module 5

Lecture 1. Equivalence of sequences and functions

Lecture 2. Angle between Functions & their Decomposition

Demo: Additional Information on Direct-Sum

Module 6

Lecture 1.Introduction to filter banks

Lecture 2.Haar Analysis Filter Bank in Z-domain

Lecture 3.Haar Synthesis Filter Bank in Z-domain.

Module 7

Lecture 1.Moving from Z-domain to frequency domain

Lecture 2.Frequency Response of Haar Analysis Low pass Filter bank

Lecture 3.Frequency Response of Haar Analysis High pass Filter bank

Module 8

Lecture 1.Ideal two-band filter bank

Lecture 2.Disqualification of Ideal filter bank

Lecture 3.Realizable two-band filter bank

Demo: Demonstration: DWT of images

Module 9

Lecture 1.Relating Fourier transform of scaling function to filter bank

Lecture 2.Fourier transform of scaling function

Lecture 3.Construction of scaling and wavelet functions from filter bank

Demo: Demonstration: Constructing scaling and wavelet functions.

Module 10

Lecture 1.Introduction to upsampling and down sampling as Multirate operations

Lecture 2.Up sampling by a general factor M- a Z-domain analysis.

Lecture 3.Down sampling by a general factor M- a Z-domain analysis.

Lecture 2.Effect of X (-Z) in time domain and aliasing.

Module 12

Lecture 1.Revisiting aliasing and the Idea of perfect reconstruction

Lecture 2.Applying perfect reconstruction and alias cancellation on Haar MRA

Lecture 3.Introduction to Daubechies family of MRA.

Lecture 2.Applying perfect reconstruction condition to obtain filter coefficient

Module 14

Lecture 1.Effect of minimum phase requirement on filter coefficients

Lecture 2.Building compactly supported scaling functions

Lecture 3.Second member of Daubechies family.

Lecture 2.Revisiting Fourier Transform and Parseval's theorem

Lecture 3.Transform Analysis of Haar Wavelet function

Module 16

Lecture 1.Nature of Haar scaling and Wavelet functions in frequency domain

Lecture 2.The Idea of Time-Frequency Resolution.

Lecture 3.Some thoughts on Ideal time- frequency domain behavior

Week 7

Lecture 2.Defining Mean, Variance and “containment in a given domain”

Lecture 3.Example: Haar Scaling function

Lecture 4.Variance from a slightly different perspective

Module 18

Lecture 1.Signal transformations: effect on mean and variance

Lecture 2.Time-Bandwidth product and its properties.

Lecture 3.Simplification of Time-Bandwidth formulae

Module 19

Lecture 1. Introduction

Lecture 2.Evaluation of Time-Bandwidth product

Lecture 3.Optimal function in the sense of Time-Bandwidth product

Lecture 2.Journey from infinite to finite Time-Bandwidth product of Haar scaling function

Lecture 3.More insights about Time-Bandwidth product

Lecture 4.Time-frequency plane

Lecture 5.Tiling the Time-frequency plane

Module 21

Lecture 1.STFT: Conditions for valid windows

Lecture 2.STFT: Time domain and frequency domain formulations.

Lecture 3.STFT: Duality in the interpretations

Lecture 4.Continuous Wavelet Transform (CWT)

Conclusive Remarks and Future Prospects

1. Michael W. Frazier, "An Introduction to Wavelets through Linear Algebra”, Springer, 1999.

2. Stephane Mallat, "A Wavelet Tour of Signal Processing", Academic Press, Elsevier, 1998, 1999, Second Edition.

3. http://nptel.ac.in/courses/117101001/: The lecture series on Wavelets and Multirate Digital Signal Processing created by Prof. Vikram M. Gadre in NPTEL.

4. Barbara Burke Hubbard, "The World according to Wavelets - A Story of a Mathematical Technique in the making", Second edition, Universities Press (Private) India Limited 2003.

5. P.P. Vaidyanathan, "Multirate Systems and Filter Banks", Pearson Education, Low Price Edition.

Certification exam:

- The exam is optional for a fee. Exams will be on 23 April, 2017.
- Time: Shift 1: 9 AM-12 Noon; Shift 2: 2 PM-5 PM
- Any one shift can be chosen to write the exam for a course.
- Registration URL: Announcements will be made when the registration form is open for registrations.
- The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published.

**Certificate:**

- Final score will be calculated as : 25% assignment score + 75% final exam score
- 25% assignment score is calculated as 25% of average of 8 weeks course: Best 6 out of 8 assignments.
- E-Certificate will be given to those who register and write the exam and score greater than or equal to 40% final score. Certificate will have your name, photograph and the score in the final exam with the breakup. It will have the logos of NPTEL and IIT Bombay . It will be e-verifiable at nptel.ac.in/noc.